Geometry of curves and surfaces through the contact map

P.J. Giblin∗ S. Janeczko

Abstract

We introduce a new approach to the study of affine equidistants and centre symmetry sets via afamily of maps obtained by reflexion in the midpoints of chords of a submanifold of affine space. Weapply this to surfaces in R

3, previously studied by Giblin and Zakalyukin, and then apply the sameideas to surfaces in R

4, elucidating some of the connexions between their geometry and the family ofreflexion maps. We also point out some connexions with symplectic topology.

1 Introduction

A number of recent works have studied curves and surfaces in affine space from the point of view of abifurcation of ‘central symmetry’. Some of these concentrate on purely local, or multi-local results—depending on small neighbourhoods of two or more points [6, 7, 8, 12]—while others involve moreglobal, topological properties such as the total number of singular points [3, 5]. A closed manifold M ,such as a planar ellipse or an ellipsoid in 3-space, which is globally symmetric about a point O—whichhas, in other words, global central symmetry—is invariant under reflexion through O. Given p ∈ M ,we can take a neighbourhood U of p in M : its reflexion in O coincides exactly with a neighbourhood ofthe reflected point p′ ∈ M . When M no longer has a global centre O we can still consider a chord pp′

joining points p, p′ of M and reflect the neighbourhood U in the midpoint of this chord: instead of thereflected neighbourhood exactly coinciding with a neighbourhood of p′ there will be some measurablecontact between these two. It turns out that this simple construction has very close connections withprevious work on bifurcations of central symmetry, helps to explain some of the results of that work ina simple geometrical way, and opens up a number of interesting avenues for further investigation. Itis the purpose of this article to introduce the ‘contact’ approach, and to give some preliminary resultsand applications.

Previous work in this area has concentrated on two basic affine constructions, both of which involvethe consideration of chords pp′ of M for which the tangent lines or planes at the endpoints p and p′

are parallel. We shall relax this condition below, but to see how it arises at all see Figure 1, left.The organization is as follows: In §2 we motivate the new approach by considering a case studied

in some detail in [7], namely that of surfaces in affine space R3. From §3 we apply this approach to

a new application, that of a surface M in affine space R4: the contact map in §3, the midpoint map

which associates to any pair of points of M their midpoint in R4 in §4 and finally in §5 we study

‘vanishing chords’, asking how the geometry of M at p is related to the contact map for arbitrarilysmall chords close to p.

2 The contact map for surfaces in R3.

Consider a surface M given locally by z = f(x, y) in R3, and take M in Monge form, that is f and its

first derivatives with respect to x and y vanish at (0, 0).For given points p = (s, t, f(s, t)) and p′ = (u, v, f(u, v)) on M we reflect a neighbourhood of p

on the surface M in the midpoint of the chord joining p and p′, and then write down the contactbetween this reflected surface, which passes through p′, and M itself at p′. The midpoint of the chord

∗Peter Giblin gratefully acknowledges the support of the Center for Advanced Studies at the Warsaw University of

Technology during October 2009.

1

E

E

p

p 'q

q '

013

2 4

M

M

E

Figure 1: Left: E is the ‘half-way equidistant’, having two components, for the two-lobed curve M : it is the locusof midpoints of parallel tangent chords of M such as pp′ and qq′. Points outside M and E are the midpoints of zerochords of M , and crossing M this number changes by 1 while crossing E it changes by 2. (Compare [4].) The unionM ∪ E is the bifurcation set of the count of midpoints. Right: for this (convex) M many equidistants are drawn:each one is the locus of points at a fixed ratio of distance along a parallel tangent chord, the half-way equidistantE being drawn heavily. The cusps on the equidistants trace out the centre symmetry set (CSS), also known as theMinkowski set. The CSS is also the envelope of parallel tangent chords.

is(

12 (s + u), 1

2 (t + v), 12 (f(s, t) + f(u, v))

), and reflecting the point (s+X, t+Y, f(s+X, t+Y )), close

to p, in this midpoint gives

(x, y, z) = (u − X, v − Y, f(s, t) + f(u, v) − f(X + s, Y + t)) . (1)

For fixed s, t, u, v this parametrizes the reflected surface close to p′. To find the contact with M at p′

we write down an equation for M , namely f(x, y) − z = 0 and substitute for x, y, z from (1). Thisgives the contact function

F(s,t,u,v) : (R2,0) → (R, 0), F(s,t,u,v)(X, Y ) = f(X +s, Y + t)+f(u−X, v−Y )−f(s, t)−f(u, v). (2)

Because we are measuring contact it is strictly the contact class of F at X = Y = 0 which we need,but for functions this is the same as the right-equivalence class, in particular there are two normalforms for A2k−1, namely X2 ± Y 2k. Since (using subscripts without brackets for partial derivatives)

FX(0, 0) = fx(s, t) − fx(u, v), FY (0, 0) = fy(s, t) − fy(u, v), (3)

it is clear that F is singular at (0, 0) if and only if the tangent planes to M at p and p′ are parallel.We shall also be interested in s, t, u, v all tending to 0, and we shall ask which singularities of

the contact function can persist as the chords shrink to zero, referring to this as the singularitiesof vanishing chords: note that the chords need to be non-trivial, that is they need to have distinctendpoints, so either s 6= u or t 6= v. Thus we are considering chords whose midpoints lie very close toM ; compare Figure 1.

Note that we can also put s = t = u = v = 0 in the family, giving

F0(X, Y ) = f(X, Y ) + f(−X,−Y ). (4)

This function measures the contact at the origin between M and the reflexion of M through the origin.It has only terms of even degree in X and Y . Clearly, F0 is singular at (0, 0), that is both F0X(0, 0)and F0Y (0, 0) are zero there. Also the hessian of F0 is zero at the origin, that is F0 has type A3 atleast, if and only if fxxfyy = f2

xy there, that is to say if and only if the origin is parabolic on M . There

are indeed two cases, according as F0 has type A+3 or A−

3 . In the first case the intersection of the twosurfaces is a single point while in the latter case it is two curves with ordinary tangency. See Figure 2,left, for the A−

3 case.Assuming then that the origin is parabolic, we can take f in the form

f(x, y) = f20x2 + f30x

3 + f21x2y + f12xy2 + f03y

3 + f40x4 + . . . + f04y

4 + . . . (f20 6= 0), (5)

2

Figure 2: Left: a surface and its reflexion in the origin having A−

3 contact. Right: A−

5 contact.

E

M M

P a r a b o l i c C u r v e

H a l f C u s p E d g e

E

Figure 3: Left: the half-way equidistant E at ordinary parabolic points is a smooth surface with boundary alongthe parabolic curve. Right: at an A∗

2 point the half-way equidistant acquires a ‘half-cuspidal edge’. This modeldoes not show the surface and the parabolic set has become a straight line.

and then F0 omits all the odd degree terms (and doubles the even degree terms), so that F0 has typeA5 at least if and only if f04 = 0. Provided the origin is not a cusp of Gauss (that is, provided f03 6= 0),the condition f04 = 0 is the condition for what is called in [8] an ‘A∗

2 point’. (Note that the A2 hererefers to a singularity of the height function at the origin, not to the function F0 ! Parabolic pointsare ‘A2-points’ in this sense and cusps of Gauss are ‘A3-points’.)

Proposition 2.1 If the origin is a non-parabolic point of M , then F0 is a Morse function. A parabolicpoint, not a cusp of Gauss, is of type A∗

2 only if F0 has type A5.

Remarks 2.2 1. The last condition in the Proposition is not quite ‘if and only if’. This is becausethe precise condition for an A∗

2 point, as in [8], requires certain open conditions on the odd degreeterms of the expansion of f which of course cannot affect the type of F0 at the origin. However theopen conditions on the even degree terms of f which are required by an A∗

2 point guarantee that thecontact is of type A5 exactly. Also, while the contact can have type A+

5 or A−

5 , no such distinctionappears in [8].

2. The special A∗2 parabolic points have a significant impact on the half-way equidistant—the locus

of midpoints of parallel tangent chords—as detailed in [14, 8]. See Figure 3 for an illustration of thisdistinction. However Proposition 2.1 appears to be the first simple interpretation of the special natureof these points, which are not detected by the usual flat geometry of a surface in R

3.

3

2.1 Vanishing chords for surfaces in R3

Now let us turn to the family of functions F in (2). As above, (3), F is singular at (X, Y ) = (0, 0)if and only if the tangent planes to M at the points with parameters (s, t) and (u, v) are parallel. Inparticular, if s = u and t = v then these partial derivatives are always zero. Eliminating these ‘trivialchords’ is a familiar problem of ‘avoiding the diagonal’. To help in this we shall change coordinates inF by

s +u = 2a, s−u = 2b, t + v = 2c, t− v = 2d, so that s = a + b, u = a− b, t = c + d, v = c− d; (6)

the diagonal s = u, t = v then becomes b = d = 0. We continue to use F for the resulting family,

F(a,b,c,d)(X, Y ) = f(X + a + b, Y + c + d) + f(a− b − X, c− d − Y ) − f(a + b, c + d) − f(a− b, c− d).

The objective in what follows is to determine which singularities of F(a,b,c,d) at X = Y = 0 occurfor pairs of distinct points (that is, b, d not both zero) arbitrarily close to (a, b, c, d) = 0. So weare interested in the contact singularities corresponding to families of vanishing chords, with distinctendpoints (‘nontrivial chords’), close to the given point, the origin, on M . The answer, as withProposition 2.1 for the function F0, will depend on the geometry of M at the origin.

Consider first the map

G : R4,0 → R

2,0 given by (a, b, c, d) 7→

(∂F

∂X(0, 0),

∂F

∂Y(0, 0)

).

The jacobian matrix of this map at (a, b, c, d) = 0 is(

0 4f20 0 2f11

0 2f11 0 4f02

).

It follows that, if the origin is not parabolic, then this matrix has rank 2 and furthermore b, d can beexpressed uniquely as functions of a and c close to the origin. But we know that G(a, 0, c, 0) ≡ 0 sothese unique functions are in fact b = 0, d = 0. Thus:

Proposition 2.3 If the origin is a non-parabolic point of M , there is a neighbourhood of (a, b, c, d) = 0

in which F(a,b,c,d) is singular at (X, Y ) = (0, 0) if and only if the chord is trivial: b = d = 0.

Thus at a non-parabolic point there are no families of nontrivial vanishing chords giving singularcontact maps at all. In a sense this is ‘obvious’ since it says that sufficiently close to a non-parabolicpoint there are no pairs of distinct points with parallel tangent planes. But note that excluding thediagonal b = d = 0 was a non-trivial step.

Suppose now that the origin is a parabolic point of M , and write f as in (5). The condition forFX = ∂F/∂X at (X, Y ) = (0, 0) to be zero has an expansion about the origin in a, b, c, d-space ofthe form 0 = 4f20b + higher order terms, so that this condition can be solved uniquely for b as asmooth function b = B(a, c, d). But we know that b = d = 0 identically satisfies this condition, soB(a, c, d) = dB1(a, c, d) for a smooth function B1. Substituting this solution for b into the conditionfor FY = ∂F/∂Y at (X, Y ) = (0, 0) the result must be divisible by d since we know that d = 0 (whichimplies b = 0 now) identically satisfies this condition. We find in fact that the condition, divided byd, takes the form 0 = 4f12a + 12f03c + higher terms.

Assume now that the origin is not a cusp of Gauss on M . Then f03 6= 0 and we can solve the lastcondition for c as a smooth function of a, d, and hence also b as such a function. Let us write

b = dB1(a, d) = d

((f2

12 − 3f03f21)a

3f20f03+ h.o.t.

), c = C(a, d) = −

f12

3f03a + h.o.t.,

as the unique solutions obtained in this way. Thus the solutions of ∂F/∂X = ∂F/∂Y = 0 at (X, Y ) =(0, 0) take the form

(i) (a, 0, c, 0), or (ii) (a, dB1(a, d), C(a, d), d), intersecting in points (iii) (a, 0, C(a, 0), 0). (7)

The set of points (ii) when d 6= 0 consists of the ‘genuine’ pairs of distinct points—that is, nontrivialchords—where tangent planes are parallel; as d → 0 these tend to pairs of repeated points, which areparametrized by (iii). Thus (iii) represents the parabolic curve on M ; more precisely (a, C(a, 0)) is aparametrization of the parabolic curve on M .

4

Proposition 2.4 At a parabolic point O of M , not a cusp of Gauss, the nontrivial vanishing chordsgiving singular contact maps are those with a, b, c, d as in (7)(ii).

We now go on to consider the condition for F to have a non-Morse singularity at X = Y = 0, forpoints (a, b, c, d) of type (7)(ii). Let

H(a, b, c, d) = (FXXFY Y − F 2XY )(0, 0), and H(a, d) = H(a, dB1(a, d), C(a, d), d).

Now H(a, 0) evaluates H at (a, 0, C(a, 0), 0) and from the definition of F , evaluating at X = Y = 0,

FXX = fxx(a + b, c + d) + fxx(a − b, c − d), FXY = fxy(a + b, c + d) + fxy(a − b, c − d),

FY Y = fyy(a + b, c + d) + fyy(a − b, c − d).

Consequently, putting b = d = 0, we find that H(a, 0) = 4(fxxfyy − f2xy), evaluated at (a, C(a, 0)),

and this is zero since, as remarked above, (a, C(a, 0)) is the parabolic curve on M . Thus H(a, 0) isidentically 0 and this shows H is divisible by d.

We can go on to show that H is divisible by d2. The definitions of B1 and C are that, at X = Y = 0,

FX(a, dB1(a, d), C(a, d), d) ≡ 0, FY (a, dB1(a, d), C(a, d), d) ≡ 0,

and these equations can be differentiated with respect to d (or indeed a) to obtain other identities.Careful inspection of ∂B1/∂d and ∂C/∂d shows that these are identically zero when evaluated at(a, C(a, 0)). It follows by a direct calculation that ∂H/∂d(a, 0) is identically zero, which now im-plies that H(a, d) is divisible by d2: H(a, d) = d2H1(a, d) for a smooth function H1. Furthermore,calculation shows that

H(a, d) = d2(64f04f20 + h.o.t.).

This now allows us to avoid the diagonal, since that is represented here by d = 0. We conclude that,when the origin is a parabolic point and not a cusp of Gauss on M , and when f04 6= 0, which meansthat the origin is not an A∗

2 point in the notation of [8], there are no nontrivial vanishing chords forwhich the singularity of F is non-Morse. Hence we have another geometrical interpretation of A∗

2

points:

Proposition 2.5 Suppose that O is a parabolic point, not a cusp of Gauss, on M . Then the exis-tence of nontrivial chords arbitrarily close to O (vanishing chords) for which the contact singularity isdegenerate (non-Morse) implies that O is an A∗

2 point of M .

Remark 2.6 We shall not pursue the above any further than A∗2 points. It is clearly of interest

to compare higher singularities of the contact function for vanishing chords with the classification ofequidistants undertaken in [8]. This will be done elsewhere.

3 The contact map F for surfaces in R4

We now turn to a new situation, that of a surface in affine space R4. Much of the above analysis can

also be carried out in this situation and we shall summarize the results here. Our references for theflat geometry of surfaces in R

4 are [1, 11].Because of the additional complications as compared with R

3 we start with the simplest situationof two surface patches M, N and points a0 = (−1, 0, 0, 0) ∈ M, b0 = (1, 0, 0, 0) ∈ N , with midpointof the chord a0b0 at the origin. As in §2 we consider the contact between the reflexion of M in thismid-point and N at b0. Let the surfaces be in the following form:

M : γ(s, t) = (p(s, t) − 1, q(s, t), s, t), N : δ(u, v) = (f(u, v) + 1, g(u, v), u, v), (8)

where s, t; u, v are small, and zero at a0, b0 respectively.The reflexion of M in the origin consists of points (−p(s, t) + 1,−q(s, t),−s,−t) and to measure

the contact of this with N at b0 we consider the composite (see [10])

R2 → R

4 → R2

(s, t) → (−p + 1,−q,−s,−t)(w, x, y, z) → (f(y, z) + 1 − w, g(y, z) − x)

5

This composite is the contact map for the chord a0b0:

F : R2 → R

2 where F (s, t) = (p(s, t) + f(−s,−t), q(s, t) + g(−s,−t)). (9)

It should be noted that, since we are measuring contact between two surfaces in R4, it is the contact

class of this map which is relevant. In particular the corank 1 singularities will be with normal forms(x, y) 7→ (x, y2) (fold), (x, y3) (cusp), (x, y4) (swallowtail). We shall not follow up these cases hereexcept to note two propositions (3.1 and 3.2), the first of which is a straightforward calculation.

Proposition 3.1 The contact map is singular at a0 if and only the tangent planes at a0 and b0 are‘weakly parallel’, that is have a nonzero tangent vector in common.

3.1 The generating function and its caustic, the CSS

The generating function, as in [6], is the following. Consider two surfaces M, N as in (8). Then thegenerating function is

F(s, t, u, v, P, λ, Q) = λ〈γ(s, t) − Q, P 〉 + (1 − λ)〈δ(u, v) − Q, P 〉,

where P is a direction in R4, Q ∈ R

4 and 〈 〉 can be interpreted as scalar product or as evaluationof a covector on a vector. For a given λ, the equidistant is the set of points Q = (Q1, Q2, Q3, Q4) ofthe form Q = λa + (1 − λ)b where a, b are points of M, N respectively at which the tangent planesare weakly parallel. See Figure 1 for an illustration in R

2. As before, λ and 1 − λ give the sameequidistant, and λ = 1

2 is then special. The family of equidistants is precisely the set of points

(λ, Q) : F =

∂F

∂s=

∂F

∂t=

∂F

∂u=

∂F

∂v=

∂F

∂P= 0

,

where the last equation represents three equations. The common normal to M and N when s =t = u = v = 0 is (−qt(0, 0), pt(0, 0), 0, 0) = (−q01, p01, 0, 0), in the notation of §3, so we take P =(−(q01/p01 + P1, 1, P3, P4) as a local parametrization of P for s, t, u, v small, assuming, as before thatp01 6= 0. The ‘base value’ of Q1 is given by F = 0 and is (1−2λ, 0, 0, 0). The base values of Q2, Q3, Q4

are 0 since the point Q always lies on the chord.To find the envelope point(s) (Centre Symmetry Set, or CSS point(s)) on the chord a0b0 we need to

impose the additional condition that the matrix of second partial derivatives of F is singular. Write pij

for the coefficient of sitj in the Taylor expansion of p at (0,0), and similarly for q, f, g. The determinantof this matrix, evaluated at s = t = u = v = P1 = P3 = P4 = 0, Q1 = 1 − 2λ, Q2 = Q3 = Q4 = 0comes to

2(1 − λ)3λ3p01(λ(f20q01 − g20p01 − p20q01 + q20p01) − q20p01 + p20q01).

Thus apart from λ = 0, 1, which always give CSS points (they are the points of M, N) there is aunique, always real (but possibly infinite), value of λ on the envelope of chords. The value of λ is

λ =q20p01 − p20q01

f20q01 − g20p01 − p20q01 + q20p01, i.e. 1 −

1

λ=

f20q01 − g20p01

p20q01 − q20p01. (10)

The condition for the CSS point to be at the midpoint of the chord is that λ = 12 and this comes

tog20p01 − q01p20 − q01f20 + q20p01 = 0,

which, by calculation, is precisely the same as the cusp condition on the contact map. Hence the CSSis closely related to contact:

Proposition 3.2 The contact map has the type of a cusp (or worse) if and only if the (unique) CSSpoint on the chord is at the midpoint of the chord.

6

Remark 3.3 The generating family F defines the fibre homogeneous Lagrangian variety LF in T ∗R

4,which projects onto critical values of the λ-point map mλ and is described by equations:

∂F

∂s= 0 : λ

⟨∂γ

∂s, P

⟩= 0;

∂F

∂t= 0 : λ

⟨∂γ

∂t, P

⟩= 0;

∂F

∂u= 0 : (1 − λ)

⟨∂δ

∂u, P

⟩= 0;

∂F

∂v= 0 : (1 − λ)

⟨∂δ

∂v, P

⟩= 0;

∂F

∂P= 0 : Q = λ(γ − δ) + δ,

and the cotangent bundle lifting, (Q, Q) ∈ T ∗R

4,

Q =∂F

∂Q= −P.

The first four equations define the critical set Cm of λ-point map, i.e. Cm, given by,

det

∣∣∣∣∂γ

∂s,∂γ

∂t,∂δ

∂u,∂δ

∂v

∣∣∣∣ = 0 (11)

and P belongs to the co-normal bundle of Σm = mλ(Cm) and is homogeneous. After projectivizationT ∗

R4 → PT ∗

R4 we get a Legendrian variety which is smooth on the maximal strata of Σm and has

singularities along SingΣm.Singularities of the Legendrian projection of PLF are defined by second derivatives of F and finally

the equation

det

(B At

A 0

)= (detA)2 = 0,

and

A =

(λ

∂γ

∂s, λ

∂γ

∂t, (1 − λ)

∂δ

∂u, (1 − λ)

∂δ

∂v

)

which on the basis of (11) says that there are no other singularities of Legendre projection besides Σm.

4 The midpoint map

There is a construction which is closely related to the contact map described in §3. To start with theplanar case, that of a simple closed smooth curve M in the affine plane R

2 parametrized by γ, say,consider the map m which associates to a pair (s, t) of parameter values the midpoint of the chord,12 (γ(s) + γ(t)). This is a map R

2 → R2, and its singularities are closely connected with the half-way

equidistant, the centre symmetry set (envelope of parallel tangent chords) and the contact map. Letus consider just the local map m at a point (s, t), s 6= t. Then it is easy to check the following.

Proposition 4.1 (i) The midpoint map m is singular at (s, t) if and only if the tangents to the curveM are parallel at γ(s) and γ(t), so that the set of critical values is the half-way equidistant;(ii) m has a cusp singularity (or worse) if and only if in addition to (i) the (unique) CSS point on thischord is at the midpoint;(iii) m has a swallowtail singularity (or worse) if and only if in addition to (ii) the CSS is singular atthe midpoint;(iv) m has a lips/beaks singularity (or worse) if and only if in addition to (i) both γ(s) and γ(t) areinflexion points of M .

In the above situation, unlike that of §3, swallowtail and lips/beaks arise from distinct phenomena(iii) and (iv). Note that these will occur generically only in a 1-parameter family of curves.

For a surface M in R4, the midpoint map is a map m : M × M → R

4, thus locally a mapm : R

4 → R4. It is closely related to the contact map; for example writing down the Jacobian matrix

of m it is clear that the corank of m and the corank of the contact map, as defined locally in (9), areequal and in particular must be 0, 1 or 2. Furthermore it is not difficult to check the following:

7

Proposition 4.2 (i) For a surface in R4 the critical set Σm of m is the set of pairs of points of M

at which the tangent planes are weakly parallel; the critical values form the half-way equidistant.(ii) The condition for m to have a singularity of type Σ1,1 (or worse), that is for m restricted toΣm to be singular, is the same as the condition that the contact map have a cusp singularity; seeProposition 3.2.(iii) When m has exactly type Σ1,1 the image of the critical set of m, that is the half-way equidistant,is a 3-dimensional cuspidal edge. Its singular set is smooth, and, using the setup as in (8), its slicesby hyperplanes given by the third coordinate in R

4 equal to a small constant are cuspidal edges in R3.

Remark 4.3 The affine midpoint map is relevant for the construction of the semiclassical Wignerfunction on symplectic space which completely describes a quantum state in the classical limit. In thiscase we deal with a Lagrangian surface L in R

4 endowed with the canonical symplectic structure inDarboux form ω = dy1 ∧ dx1 + dy2 ∧ dx2, ω |L= 0 (see [2, p. 130]).

Now the two pieces M, N of generic L are generated by two independent generating functions GM ,and GN (resp.),

M : γ(s, t) =

(∂GM

∂s− 1,

∂GM

∂t, s, t

), γ∗ω = 0

N : δ(u, v) =

(∂GN

∂u+ 1,

∂GN

∂v, u, v

), δ∗ω = 0.

It was shown in [3] that the midpoint map mL lifts to a lagrangian map into cotangent bundle T ∗R

4

and its singularities are singularities of Lagrangian projection, i.e. the following diagram commutes

mL

π

R2 × R

2 (R4, ω)-

(T ∗R

4, Ω)

*ML

?

where Ω is a Liouville symplectic structure of T ∗R

4, M∗

LΩ = 0 and a generating Morse family ofML reads as follows

F (x, y, α) = GM (x − α) − GN (x + α) − αy,

where α ∈ R2 is a Morse parameter.

5 Vanishing chords for surfaces in R4

As in §2.1 we consider a neighbourhood of a point, say the origin O, on a surface N in R4. For

chords whose distinct endpoints both tend to O we ask which singularities can occur for the contactmap, defined relative to the chords as in §3. For a surface in R

3 we could identify specific geometricalrestrictions on N which allow such vanishing chords with associated singularities of different types.The types in question for a surface in R

4 are fold, cusp and swallowtail. We cover here the first twoof these, and leave the third one for discussion elsewhere.

Let us start with a surface N in ‘Monge form’

N : δ(x, y) = (f(x, y), g(x, y), x, y)

where f and g have no constant or linear terms. Specifically,

f(x, y) = f20x2 + f11xy + f02y

2 + . . . , g(x, y) = g20x2 + g11xy + g02y

2 + . . . , (12)

the coefficient of xiyj being in general fij or gij .

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We then choose two points δ(s, t) and δ(u, v) and reflect the surface N in the mid-point of thechord joining these two points. We write down the contact function between the two surfaces at thepoint δ(s, t) say (we could equally well do it at the other point, δ(u, v)). For fixed s, t, u, v the contactfunction is, for (x, y) close to (s, t),

(x, y) 7→ (f(u, v)+f(s, t)−f(x, y)−f(u+s−x, v+t−y), g(u, v)+g(s, t)−g(x, y)−g(u+s−x, v+t−y)).

We want to evaluate this at x = s, y = t so substitute X = x − s, Y = y − t and the function becomes

F (X, Y ) = (f(u, v)+f(s, t)−f(X+s, Y +t)−f(u−X, v−Y ), g(u, v)+g(s, t)−g(X+s, Y +t)−g(u−X, v−Y )),

and we are interested in the singularity of this at X = Y = 0. In particular we want to know whether,for s, t, u, v all tending to 0, F can have, say, a fold singularity, or a cusp, or some higher singularity.Thus we need to write down conditions for F to have one of these singularities, as a function of s, t, u, vand of course the Monge coefficients in the functions f and g.

Remark 5.1 Note that F is a 4-parameter family of functions R2 → R

2, parametrized by s, t, u, v.As in §2 we can consider the function F0 with s = t = u = v = 0:

F0(X, Y ) = (−f(X, Y ) − f(−X,−Y ), −g(X, Y ) − g(−X,−Y )),

which has corank 2, starting with quadratic terms in both components. Also only even degree termsin the power series expansions of f, g will appear. This map F0 can be interpreted as the contactbetween N and its reflexion through the origin.

In fact the contact type of this function is indeed related to the flat geometry of the surface N atO, giving different types for a hyperbolic point, a parabolic point, and a special parabolic point verysimilar to the A∗

2 points which arose in §2. But here we shall concentrate on the vanishing chords andtheir contact singularities.

We shall, as in §2, use the substitution in (6) to rewrite F as a function, still called F , ofa, b, c, d, X, Y . Thus

F(a,b,c,d)(X, Y ) =

(f(a + b, c + d) + f(a − b, c − d) − f(X + a + b, Y + c + d) − f(a − b − X, c − d − Y ),

g(a + b, c + d) + g(a − b, c − d) − g(X + a + b, Y + c + d) − g(a − b − X, c − d − Y )) .

5.1 Fold of F for s, t, u, v, or a, b, c, d → 0

The condition for the contact map F above to be singular—at least a fold—at X = Y = 0 is thatits Jacobian determinant J is singular there. This gives a condition say G(a, b, c, d) = 0. The lowestdegree terms of G come to

G2(a, b, c, d) = 8((f20g11 − f11g20)b2 + 2(f20g02 − f02g20)bd + (f11g02 − f02g11)d

2) (13)

= αb2 + 2βbd + γd2, say.

The key observation here is that, writing F = (F1, F2) in terms of a, b, c, d,

∂F1

∂X

∣∣∣∣X=Y =0

= −fx(a + b, c + d) + fx(a − b, c − d),

which is identically zero when b = d = 0, so that this function can be written as bB + cD where B, Dare functions of a, b, c, d. The same applies to the other three partial derivatives F1Y = ∂F1/∂Y, F2X =∂F2/∂X, F2Y = ∂F2/∂Y and therefore G can be expressed as

G(a, b, c, d) = b2(α + α1(a, b, c, d)) + 2bd(β + β1(a, b, c, d)) + d2(γ + γ1(a, b, c, d)), (14)

for smooth functions α1, β1, γ1 vanishing at a = b = c = d = 0, that is without constant terms.

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The condition for G2 = 0 (13) to have real solutions other than b = d = 0 is that its discriminantis > 0, namely

β2 > αγ, i.e. (f20g02 − f02g20)2 > (f20g11 − f11g20)(f11g02 − f02g11),

and from [1] this is exactly the condition for the origin on N to be hyperbolic. Suppose then thatthe origin is an elliptic point of N , which means that β2 < αγ. The only real solutions to G2 = 0are b = d = 0. Choose a neighbourhood U of the origin in a, b, c, d-space small enough that, for all(a, b, c, d) ∈ U ,

(β + β1(a, b, c, d))2 < 4(α + α1(a, b, c, d))(γ + γ1(a, b, c, d)).

Then every solution to G(a, b, c, d) = 0 for (a, b, c, d) ∈ U must have b = d = 0. Hence

Proposition 5.2 Suppose that the origin O is an elliptic point of N . Then there is a neighbourhoodof O such that no nontrivial chords both of whose (distinct) endpoints lie in that neighbourhood give asingular contact function. In other words: there are no nontrivial vanishing chords giving a singularcontact function. Or again: there are no nontrivial vanishing chords at whose ends the tangent planesto N are pseudo-parallel.

Suppose now that the origin O is a hyperbolic or parabolic point of N , which amounts to β2 ≥ αγin the notation above. There will now be two real solutions to G2 = 0 (13) in the b, d plane, and it isto be expected that these extend to solutions (a, b, c, d) through a = b = c = d = 0. In this case, theMorse lemma with parameters establishes that there are two families of solutions, with say a, c, d allfunctions of b, and that these solutions form a set in (a, b, c, d)-space which is locally diffeomorphic to(a, b, c, d) : d = ±b.

5.2 Cusp of F

We now ask the question: given a hyperbolic or parabolic point O of N , what is the condition thatthere exist vanishing nontrivial chords for which the contact function F has a cusp singularity? Thecalculations are similar to those above, and they are sketched below. Again the crucial observationis that the additional criterion for a cusp singularity, beyond that for a fold, has the form b2(...) +bd(...) + d2(...) = 0 for suitable functions of a, b, c, d in the brackets.

There is no single condition for a cusp of a map F : R2 → R

2 at the origin; rather there are twoalternative forms which hold in different circumstances. The precise condition (compare for example[13, Ch.3] or [9, 5.2]) is that the critical set of F passes through the origin, is smooth there, andis tangent, with 2-point contact, to the kernel line of the map at the origin. We shall concentratehere on the condition that the critical set is tangent to the kernel line. Using J for the Jacobiandeterminant of F(a,b,c,d)(X, Y ) the additional cusp condition, apart from open conditions, is (writingpartial derivatives as suffices)

JXF1X − JY F1Y = 0 at X = Y = 0 if F1X , F1Y are not both zero there

and similarly with F2 replacing F1 if F2X , F2Y are not both zero at X = Y = 0. At least one of theseconditions will hold since the critical set of F is smooth at the origin. Now J = F1XF2Y −F2XF1Y and,as above, at X = Y = 0, F1X(a, 0, c, 0) and the other three similar derivatives are identically zero. Wededuce that each of the possible cusp conditions, say H(a, b, c, d), has the form b2(...)+ bd(...)+d2(...).In fact calculation shows that we can write either of the cusp conditions in the form

H(a, b, c, d) = b2(A + A1(a, b, c, d)) + bd(B + B1(a, b, c, d)) + d2(C + C1(a, b, c, d)).

Here A, B, C are real numbers, giving the ‘lowest terms’ of H , so that A1, B1, C1 are smooth functionswithout constant terms. The two alternative forms of the cusp condition need the quadratic terms off , resp. g, not to be a perfect square. So we are excluding the special case where f and g are bothsquares.

Now calculation shows that, for the cusp condition requiring the quadratic terms of f not to be aperfect square, the condition that the quadratic terms of the two conditions, fold and cusp,

Ab2 + Bbd + Cd2 = 0 and αb2 + βbd + γd2 = 0 (see (13))

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share a common root other than b = d = 0 is precisely that the origin O is parabolic on the surfaceN . We can now apply an argument similar to that in §5.1 to show that, assuming the origin is notparabolic on N and a, b, c, d are sufficiently small, there will be no common solutions to G = 0 andH = 0 other than b = d = 0. Hence:

Proposition 5.3 Suppose that the origin O is a hyperbolic point but not a parabolic point of N .Suppose in addition that at least one of the quadratic parts of f and g is not a perfect square. Thenthere is a neighbourhood of O such that no nontrivial chords, both of whose (distinct) endpoints liein that neighbourhood, give a contact function with a cusp singularity. In other words: there are nonontrivial vanishing chords giving a contact function with a cusp.

It is also clear that, at a parabolic point, there will be such vanishing chords. The next case is theswallowtail, but there is also the special case where f and g are both perfect squares, which is notdistinguished in the flat geometry of the surface. These cases are under investigation.

References

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[2] A.M. Ozorio de Almeida and J.H. Hannay, ‘Geometry of two dimensional tori in phase space:projections, sections and the Wigner function’ Annals of Physics Vol. 138, (1982), 115–154.

[3] W.Domitrz and P. de M. Rios, ‘Singularities of global centre symmetry sets of Lagrangian sub-manifolds’, preprint.

[4] P.J.Giblin, ‘Affinely invariant symmetry sets’, Caustics ’06, Banach Center Publications 82 (2008),71–84.

[5] P.J.Giblin and P.A.Holtom, ‘The centre symmetry set’, Geometry and Topology of Caustics, Ba-nach Center Publications, Vol 50, ed. S.Janeczko and V.M.Zakalyukin, Warsaw, 1999, pp.91–105.

[6] P.J.Giblin and V.M.Zakalyukin, ‘Singularities of centre symmetry sets’, Proc. London Math. Soc.90 (2005), 132–166.

[7] P.J.Giblin and V.M.Zakalyukin, ‘Recognition of centre symmetry set singularities’, GeometriaeDedicata 130 (2007), 43–58.

[8] P.J.Giblin, J.P.Warder and V.M.Zakalyukin, ‘Bifurcations of affine equidistants’, Proceedings ofthe Steklov Institute of Mathematics 267 (2009), 57–75.

[9] Yung-Chen Lu, Singularity theory and an introduction to catastrophe theory, Springer-Verlag 1976.

[10] J. A. Montaldi, ‘On generic composites of maps’, Bull. Lond. Math. Soc. 23 (1991), 81–85.

[11] D. K. H. Mochida, M. C. R. Fuster and M. A. S. Ruas, ‘The geometry of surfaces in 4-space froma contact viewpoint,’ Geometria Dedicata 54 (1995), 323–332.

[12] G.Reeve and V.Zakalyukin, ‘Singularities of the Minkowski Set and Affine Equidistants for aCurve and a Surface’, preprint, University of Liverpool, 2010.

[13] Farid Tari, Some applications of singularity theory to the geometry of surfaces, PhD thesis, Uni-versity of Liverpool, 1990.

[14] J.Paul Warder, PhD thesis, University of Liverpool, 2009. Available athttp://www.liv.ac.uk/∼pjgiblin

S.Janeczko: Institute of Mathematics P.Giblin: Department of Mathematical SciencesPolish Academy of Sciences The University of Liverpoolul. Sniadeckich 8, 00-950 Warsaw, Poland, and Liverpool L69 7ZL, U.K.Faculty of Mathematics and Information ScienceWarsaw University of TechnologyPl. Politechniki 1, 00-661 Warsaw, Poland.janeczko@impan.gov.pl pjgiblin@liv.ac.uk

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